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 A306015 Exponential series expansion of (exp(x*y) + sinh(x) - cosh(x))/(1 - x). 2
 0, 1, 1, 1, 2, 1, 4, 6, 3, 1, 15, 24, 12, 4, 1, 76, 120, 60, 20, 5, 1, 455, 720, 360, 120, 30, 6, 1, 3186, 5040, 2520, 840, 210, 42, 7, 1, 25487, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 229384, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS From David Callan, Dec 18 2021: (Start) For 0 <= k <= n, T(n,k) is the number of nonderangements of size n in which k of the fixed points are colored red. In particular, with D_n the derangement number A000166(n), T(n,0) = n! - D_n. For a general example, T(3,1) = 6 counts the colored permutations R23, R32, 1R3, 3R1, 12R, 21R where the red fixed points are indicated by "R". For n >= k >= 1, T(n,k) = n!/k!. Proof. In a colored permutation, such as 3R7R516 counted by T(n,k) with n = 7 and k = 2, the R's indicate (red) fixed points and so no information is lost by rank ordering the remaining entries while retaining the placement of the R's: 2R5R314. The result is a permutation of the set consisting of 1,2,...,n-k and k R's; there are n!/k! such permutations and the process is reversible. QED. (End) LINKS G. C. Greubel, Rows n=0..99 of triangle, flattened EXAMPLE n | k = 0 1 2 3 4 5 6 7 8 9 --+---------------------------------------------------------- 0 | 0 1 | 1, 1 2 | 1, 2, 1 3 | 4, 6, 3, 1 4 | 15, 24, 12, 4, 1 5 | 76, 120, 60, 20, 5, 1 6 | 455, 720, 360, 120, 30, 6, 1 7 | 3186, 5040, 2520, 840, 210, 42, 7, 1 8 | 25487, 40320, 20160, 6720, 1680, 336, 56, 8, 1 9 | 229384, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1 MAPLE gf := (exp(x*y) + sinh(x) - cosh(x))/(1 - x): ser := series(gf, x, 16): L := [seq(n!*coeff(ser, x, n), n=0..14)]: seq(seq(coeff(L[k+1], y, n), n=0..k), k=0..12); MATHEMATICA Join[{0}, With[{nmax = 15}, CoefficientList[CoefficientList[Series[ (Exp[x*y] + Sinh[x] - Cosh[x])/(1 - x), {x, 0, nmax}, {y, 0, nmax}], x], y ]*Range[0, nmax]!] // Flatten ] (* G. C. Greubel, Jul 18 2018 *) CROSSREFS A094587 with an extra first column A002467. Row sums are A306150. Sequence in context: A257640 A262930 A293387 * A171087 A105364 A180405 Adjacent sequences: A306012 A306013 A306014 * A306016 A306017 A306018 KEYWORD nonn,tabl AUTHOR Peter Luschny, Jun 23 2018 STATUS approved

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Last modified June 5 19:06 EDT 2023. Contains 363138 sequences. (Running on oeis4.)